We decribe a number of properties of a median metric space. In particular, we show that a complete connected median metric of finite rank admits a canonical bilipschiz equivalent CAT(0) metric. Metric spaces of this sort arise, up to bilipschitz equivalence, as asymptotic cones of certain classes of finitely generated groups, and the existence of such a structure has various consequences for the large scale geometry of the group.
Groups, Geom. Dyn. 10 (2016) 279--317.